Theorem cvlatcvr1 29531

Description: An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012.)

Hypotheses

Ref	Expression
cvlatcvr1.j	|- .\/ = ( join ` K )
cvlatcvr1.c	|- C = ( <o ` K )
cvlatcvr1.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
cvlatcvr1	|- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> P C ( P .\/ Q ) ) )

Proof of Theorem cvlatcvr1

Step	Hyp	Ref	Expression
1		simp13 987	. . . 4 |- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> K e. CvLat )
2		cvlatl 29515	. . . 4 |- ( K e. CvLat -> K e. AtLat )
3	1, 2	syl 15	. . 3 |- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> K e. AtLat )
4		eqid 2283	. . . 4 |- ( meet ` K ) = ( meet ` K )
5		eqid 2283	. . . 4 |- ( 0. ` K ) = ( 0. ` K )
6		cvlatcvr1.a	. . . 4 |- A = ( Atoms ` K )
7	4, 5, 6	atnem0 29508	. . 3 |- ( ( K e. AtLat /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> ( P ( meet ` K ) Q ) = ( 0. ` K ) ) )
8	3, 7	syld3an1 1228	. 2 |- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> ( P ( meet ` K ) Q ) = ( 0. ` K ) ) )
9		eqid 2283	. . . 4 |- ( Base ` K ) = ( Base ` K )
10	9, 6	atbase 29479	. . 3 |- ( P e. A -> P e. ( Base ` K ) )
11		cvlatcvr1.j	. . . 4 |- .\/ = ( join ` K )
12		cvlatcvr1.c	. . . 4 |- C = ( <o ` K )
13	9, 11, 4, 5, 12, 6	cvlcvrp 29530	. . 3 |- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. ( Base ` K ) /\ Q e. A ) -> ( ( P ( meet ` K ) Q ) = ( 0. ` K ) <-> P C ( P .\/ Q ) ) )
14	10, 13	syl3an2 1216	. 2 |- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> ( ( P ( meet ` K ) Q ) = ( 0. ` K ) <-> P C ( P .\/ Q ) ) )
15	8, 14	bitrd 244	1 |- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> P C ( P .\/ Q ) ) )

Syntax hints:   -> wi 4   <-> wb 176   /\ w3a 934   = wceq 1623   e. wcel 1684   =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   joincjn 14078   meetcmee 14079   0.cp0 14143   CLatccla 14213   OMLcoml 29365   <o ccvr 29452   Atomscatm 29453   AtLatcal 29454   CvLatclc 29455

This theorem is referenced by:  cvlatcvr2  29532  atcvr1  29606

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512